Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that

uv = vu = 1_R, where 1_R is the multiplicative identity element.^[1][2]

The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the trivial ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.

Unfortunately, the term unit is also used to refer to the identity element 1_R of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. (For this reason, some authors call 1_R "unity", and say that R is a "ring with unity" rather than "ring with a unit".)

Group of units

The units of R form a group U(R) under multiplication, the group of units of R. Other common notations for U(R) are R*,  R^×, and E(R)  (for the German term Einheit).

In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ~ on R called associatedness such that

r ~ s

means that there is a unit u with r = us.

One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f : R → S induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring construction.

In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R).

A ring R is a division ring if and only if R* = R \ {0}.

Examples

• In the ring of integers, Z, the units are ±1. The associates are pairs n and −n.
• In the ring of integers modulo n, Z/nZ, the units are the congruence classes (mod n) which are coprime to n. They constitute the multiplicative group of integers (mod n).
• Any root of unity is a unit in any unital ring R. (If r is a root of unity, and rⁿ = 1, then r⁻¹ = r^{n − 1} is also an element of R by closure under multiplication.)
• If R is the ring of integers in a number field, Dirichlet's unit theorem states that the group of units of R is a finitely generated abelian group. For example, we have (√5 + 2)(√5 − 2) = 1 in the ring of integers of Q[√5], and in fact the unit group is infinite in this case. In general, the unit group of a real quadratic field is always infinite (of rank 1).
• In the ring M(n,F) of n×n matrices over a field F, the units are exactly the invertible matrices.

References